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Sampling in quasi shift-invariant spaces and Gabor frames generated by ratios of exponential polynomials

Alexander Ulanovskii, Ilya Zlotnikov

TL;DR

The paper addresses sampling and interpolation in shift-invariant and quasi shift-invariant spaces generated by two families of entire/meromorphic generators with exponential or Gaussian decay, and connects these results to Gabor frames with semi-regular lattices. It develops Beurling-density-based criteria and stability analyses for Gamma-shifts, employing weak-limit arguments and Jensen formulas to obtain sharp density thresholds. Key contributions include explicit density conditions for stable sampling and interpolation, constructions of non-uniqueness sets at critical densities, and concrete frame results for semi-regular Gabor systems driven by generators from $\mathcal{K}(\alpha)$ and $\mathcal{C}(\alpha)$. The findings advance the understanding of irregular sampling and frame design in time-frequency analysis, with practical implications for signal reconstruction on nonuniform grids and for the design of Gabor frames with Gaussian- or exponential-polynomial generators.

Abstract

We introduce two families of generators (functions) $\mathcal{G}$ that consist of entire and meromorphic functions enjoying a certain periodicity property and contain the classical Gaussian and hyperbolic secant generators. Sharp results are proved on the density of separated sets that provide non-uniform sampling for the shift-invariant and quasi shift-invariant spaces generated by elements of these families. As an application, we obtain new sharp results on the density of semi-regular lattices for the Gabor frames generated by elements from these families.

Sampling in quasi shift-invariant spaces and Gabor frames generated by ratios of exponential polynomials

TL;DR

The paper addresses sampling and interpolation in shift-invariant and quasi shift-invariant spaces generated by two families of entire/meromorphic generators with exponential or Gaussian decay, and connects these results to Gabor frames with semi-regular lattices. It develops Beurling-density-based criteria and stability analyses for Gamma-shifts, employing weak-limit arguments and Jensen formulas to obtain sharp density thresholds. Key contributions include explicit density conditions for stable sampling and interpolation, constructions of non-uniqueness sets at critical densities, and concrete frame results for semi-regular Gabor systems driven by generators from and . The findings advance the understanding of irregular sampling and frame design in time-frequency analysis, with practical implications for signal reconstruction on nonuniform grids and for the design of Gabor frames with Gaussian- or exponential-polynomial generators.

Abstract

We introduce two families of generators (functions) that consist of entire and meromorphic functions enjoying a certain periodicity property and contain the classical Gaussian and hyperbolic secant generators. Sharp results are proved on the density of separated sets that provide non-uniform sampling for the shift-invariant and quasi shift-invariant spaces generated by elements of these families. As an application, we obtain new sharp results on the density of semi-regular lattices for the Gabor frames generated by elements from these families.
Paper Structure (27 sections, 18 theorems, 127 equations, 1 figure)

This paper contains 27 sections, 18 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Sampling sets and Beurling densities
  3. Class K(a)
  4. Class C(a)
  5. Stability of Gamma-shifts
  6. Sampling for generators from K(a) and C(a)
  7. Interpolating sets for quasi shift-invariant spaces
  8. Gabor frames
  9. Structure of the paper
  10. Preliminaries
  11. Stability of Gamma-shifts
  12. Stability of Z-shifts for generators in C(a)
  13. Stability in VpGamma(g)
  14. Uniqueness Sets
  15. Proof of Proposition \ref{['uniq_prop']}, Part (I)
  16. ...and 12 more sections

Key Result

Lemma 1.5

Assume $\mathcal{G}\in W_0$ and $p\in[1,\infty]$. Then the integer-shifts of $\mathcal{G}$ are $l^p$-stable if and only if the Fourier transform $\hat{\mathcal{G}}$ of $\mathcal{G}$ satisfies

Figures (1)

  • Figure 1: Case 2 for $N=5$

Theorems & Definitions (46)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 36 more